The .375 Winchester, like the .357 Maximum, will take deer reliably out to 200 yards with good shot placement. MOA accuracy should be no big deal with a TC with some load development, but it is far more than you need for 200 yard shooting. I don't know that the Max is a factory round any longer, so that may a problem for you. The Max, if you can find a factory loading, is a great round for deer with a much wider variety of bullets that are suitable for your intended purpose. When ranges get out to 200 yards, the pointed Hornady 180gr SSP bullet shoots a bit flatter FN's and is designed for your velocity levels. I'm not going against the .375, but the Max is a softer shooting round in the Contender, and is much more versatile as far as bullet availability.
There is no round, other than a howitzer, that gaurantees the animal will drop on the spot. If you can shoot a deer in the spine every time, it will drop on the spot. A 340 Weatherby driving a 210gr X-bullet at 3250fps will not gaurantee the animal doesn't take another step